Gamma-SAR entropy

We will see how the Shannon entropy of the Gamma-SAR model varies. It is given by \[\begin{align} H_{\Gamma_{\text{SAR}}}(L_0, \mu) &= L_0 -\ln(L_0/\mu)+\ln\Gamma(L_0)+(1-L_0)\psi^{(0)}(L_0)\\ &= \big[L_0 -\ln L_0+\ln\Gamma(L_0)+(1-L_0)\psi^{(0)}(L_0)\big] + \ln \mu. \end{align}\] where \(L_0\geq 1\) is known, and \(\mu>0\) is the mean. We see that, given \(L_0\), the entropy of a random variable following the Gamma-SAR model depends on the logarithm of the mean \(\mu\).

GI0-SAR Entropy

The Shannon entropy of the GI0-SAR model is given by

\[\begin{multline} \label{E:E-GIO} H_{\mathcal{G}_I^0}(\mu, \alpha, L_0) =\underbrace{L_0 -\ln L_0+\ln\Gamma(L_0)+(1-L_0)\psi^{(0)}(L_0) +\ln \mu}_{H_{\Gamma_{\text{SAR}}}} -\ln\Gamma(L_0-\alpha)+ (L_0-\alpha) \psi^{(0)}(L_0-\alpha)\\ -(1-\alpha)\psi^{(0)}(-\alpha)+\ln (-1-\alpha)+\ln\Gamma(-\alpha)-L_0 \end{multline}\]

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